SEQUENTIAL HOMOGENIZATION OF REACTIVE TRANSPORT IN POLYDISPERSE POROUS MEDIA

Direct numerical simulations of flow and transport in porous media are computationally prohibitive due to the disparity between the typical scale at which processes are well understood (e.g., the pore-scale) and the scale of interest (the system- or field-scale). Homogenization approaches overcome some of the difficulties of full pore-scale simulations by providing an upscaled representation of fine-scale processes. Real porous systems, e.g., rocks, pose additional challenges since they usually exhibit multimodal distributions in physical and chemical properties. Perforated domains, i.e., domains with impermeable inclusions embedded in a porous matrix, represent one such example. These hierarchical media cannot be approached by a single continuum formulation. Sequential homogenization techniques build a hierarchy of effective equations that sequentially carry the smallest scale information through the intermediate scales up to the macroscale. The advantage of sequential upscaling in handling multimodal distribution in physical and chemical properties lies in its computational efficiency compared to one-step homogenization: the information about smaller-scale heterogeneity is incorporated in the subsequent scales in terms of effective media properties. Yet, existence of one or multiple intermediate scales can significantly decrease the accuracy of multiscale formulations. We show that the accuracy of multiscale methods based on sequential upscaling is strongly influenced by a combination of geometric and dynamical scale separation conditions. In particular, we investigate under which conditions sequential homogenization of reactive solute transport in geometrically and chemically heterogeneous porous domains composed of bidisperse cylinders can accurately describe pore-scale processes. We show that under appropriate conditions, expressed in terms of Peclet and Damkohler numbers and a scales separation parameter, the sequential upscaling method has second-order accuracy. We compare sequential upscaling results with the direct solution of the fully resolved pore-scale problem.